|
In mathematics, Milnor K-theory is an invariant of fields defined by . Originally viewed as an approximation to algebraic K-theory, Milnor K-theory has turned out to be an important invariant in its own right. ==Definition== The calculation of ''K''2 of a field by Matsumoto led Milnor to the following, seemingly naive, definition of the "higher" ''K''-groups of a field ''F'': : the quotient of the tensor algebra over the integers of the multiplicative group ''F''× by the two-sided ideal generated by the elements : for ''a'' ≠ 0, 1 in ''F''. The ''n''th Milnor K-group ''K''''n''M(''F'') is the ''n''th graded piece of this graded ring; for example, ''K''0M(''F'') = Z and ''K''1M(''F'') = ''F'' *. There is a natural homomorphism : from the Milnor K-groups of a field to the Quillen K-groups, which is an isomorphism for ''n'' ≤ 2 but not for larger ''n'', in general. For nonzero elements ''a''1, ..., ''a''''n'' in ''F'', the symbol in ''K''''n''M(''F'') means the image of ''a''1 ⊗ ... ⊗ ''a''''n'' in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that = 0 in ''K''2M(''F'') for ''a'' in ''F'' − is sometimes called the Steinberg relation. The ring ''K'' *M(''F'') is graded-commutative.〔Gille & Szamuely (2006), p. 184.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Milnor K-theory」の詳細全文を読む スポンサード リンク
|